Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  undif5 Structured version   Visualization version   GIF version

Theorem undif5 30294
Description: An equality involving class union and class difference. (Contributed by Thierry Arnoux, 26-Jun-2024.)
Assertion
Ref Expression
undif5 ((𝐴𝐵) = ∅ → ((𝐴𝐵) ∖ 𝐵) = 𝐴)

Proof of Theorem undif5
StepHypRef Expression
1 difun2 4390 . 2 ((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)
2 disjdif2 4389 . 2 ((𝐴𝐵) = ∅ → (𝐴𝐵) = 𝐴)
31, 2syl5eq 2848 1 ((𝐴𝐵) = ∅ → ((𝐴𝐵) ∖ 𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  cdif 3881  cun 3882  cin 3883  c0 4246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247
This theorem is referenced by:  fressupp  30451
  Copyright terms: Public domain W3C validator